Introduction to Topology of Real Algebraic Varieties

According to a tradition going back to Hilbert, for a long time the main question concerning the topology of real algebraic curves was considered to be the determination of which isotopy types are realized by nonsingular real projective algebraic plane curves of a given degree (i.e., Problem 1.1.B above). However, as early as in 1876 F. Klein Kle-22 posed the question more broadly. He was also interested in how the isotopy type of a curve is connected to the way the set $\mathbb{R}A$ of its real points is positioned in the set $\mathbb{C}A$ of its complex points (i.e., the set of points of the complex projective plane whose homogeneous coordinates satisfy the equation defining the curve).

The set $\mathbb{C}A$ is an oriented smooth two-dimensional submanifold of the complex projective plane $\mathbb{C}P^{2}$. Its topology depends only on the degree of $A$ (in the case of nonsingular $A$). If the degree is $m$, then $\mathbb{C}A$ is a sphere with $\frac{1}{2}(m-1)(m-2)$ handles. (It will be shown in Section 2.3.) Thus the literal complex analogue of Topological Classification Problem 1.1.A is trivial.

The complex analogue of Isotopy Classification Problem 1.1.B leads also to a trivial classification: the topology of the pair $(\mathbb{C}P^{2},\mathbb{C}A)$ depends only on the degree of $A$, too. The reason for this is that the complex analogue of a more refined Rigid Isotopy Classification problem 1.7.A has a trivial solution: nonsingular complex projective curves of degree $m$ form a space $\mathbb{C}NC_{m}$ similar to $\mathbb{R}NC_{m}$ (see Section 1.7) and this space is connected, since it is the complement of the space $\mathbb{C}SC_{m}$ of singular curves in the space $\mathbb{C}C_{m}(=\mathbb{C}P^{\frac{1}{2}m(m+3)})$ of all curves of degree $m$, and $\mathbb{C}SC_{m}$ has real codimension 2 in $\mathbb{C}C_{m}$ (its complex codimension is 1).

The set $\mathbb{C}A$ of complex points of a real curve $A$ is invariant under the complex conjugation involution $conj:\mathbb{C}P^{2}\to\mathbb{C}P^{2}:(z_{0}:z_{1}:z_{2})\mapsto(\overline{z} _{0}:\overline{z}_{1}:\overline{z}_{2})$. The curve $\mathbb{R}A$ is the fixed point set of the restriction of this involution to $\mathbb{C}A$.

The real curve $\mathbb{R}A$ may divide or not divide $\mathbb{C}A$. In the first case we say that $A$ is a dividing curve or a curve of type I, in the second case we say that it is a nondividing curve or a curve of type II. In the first case $\mathbb{R}A$ divides $\mathbb{C}A$ into two connected pieces.³³Proof: the closure of tne union of a connected component of $\mathbb{C}A\smallsetminus\mathbb{R}A$ with its image under $conj$ is open and close in $\mathbb{C}A$, but $\mathbb{C}A$ is connected. The natural orientations of these two halves determine two opposite orientations on $\mathbb{R}A$ (which is their common boundary); these orientations of $\mathbb{R}A$ are called the complex orientations of the curve.

A pair of orientations opposite to each other is called a semiorientation. Thus the complex orientations of a curve of type I comprise a semiorientation. Naturally, the latter is called a complex semiorientation.

The scheme of relative location of the ovals of a curve is called the real scheme of the curve. The real scheme enhanced by the type of the curve, and, in the case of type I, also by the complex orientations, is called the complex scheme of the curve.

We say that the real scheme of a curve of degree $m$ is of type I (type II) if any curve of degree $m$ having this real scheme is a curve of type I (type II). Otherwise (i.e., if there exist curves of both types with the given real scheme), we say that the real scheme is of indeterminate type.

The division of curves into types is due to Klein Kle-22. It was Rokhlin Rok-74 who introduced the complex orientations. He introduced also the notion of complex scheme and its type Rok-78. In the eighties the point of view on the problems in the topology of real algebraic varieties was broadened so that the role of the main object passed from the set of real points, to this set together with its position in the complexification. This viewpoint was also promoted by Rokhlin.

As we will see, the notion of complex scheme is useful even from the point of view of purely real problems. In particular, the complex scheme of a curve is preserved under a rigid isotopy. Therefore if two curves have the same real scheme, but distinct complex schemes, the curves are not rigidly isotopic. The simplest example of this sort is provided by the curves of degree 5 shown in Figure 8, which are isotopic but not rigidly isotopic.

A complex projective line is homeomorphic to the two-dimensional sphere.⁴⁴I believe that this may be assumed well-known. A short explanation is that a projective line is a one-point compactification of an affine line, which, in the complex case, is homeomorphic to $\mathbb{R}^{2}$. A one-point compactification of $\mathbb{R}^{2}$ is unique up to homeomorphism and homeomorphic to $S^{2}$. The set of real points of a real projective line is homeomorphic to a circle; by the Jordan theorem it divides the complexification. Therefore a real projective line is of type I. It has a pair of complex orientations, but they do not add anything, since the real line is connected and admits only one pair of orientations opposite to each other.

The action of $conj$ on the set of complex points of a real projective line is determined from this picture by rough topological arguments. Indeed, it is not difficult to prove that any smooth involution of a two-dimensional sphere with one-dimensional (and non-empty) fixed point set is conjugate in the group of autohomeomorphisms of the sphere to the symmetry in a plane.  $(^{1})$

The set of complex points of a nonsingular plane projective conic is homeomorphic to $S^{2}$, because the stereographic projection from any point of a conic to a projective line is a homeomorphism. Certainly, an empty conic, as any real algebraic curve with empty set of real points, is of type II. The empty set cannot divide the set of complex points. For the same reasons as a line (i.e. by Jordan theorem), a real nonsingular curve of degree 2 with non-empty set of real points is of type I. Thus the real scheme $\langle 1\rangle$ of degree 2 is of type I, while the scheme $\langle 0\rangle$ is of type II for any degree.

To consider further examples, it would be useful to understand what is going on in the complex domain, when one makes a classical small perturbation (see Section 1.5).

First, consider the simplest special case: a small perturbation of the union of two real lines. Denote the lines by $L_{1}$ and $L_{2}$ and the result by $C$. As we saw above, $\mathbb{C}L_{i}$ and $\mathbb{C}C$ are homeomorphic to $S^{2}$. The spheres $\mathbb{C}L_{1}$ and $\mathbb{C}L_{2}$ intersect each other at a single point. By the complex version of the implicit function theorem, $\mathbb{C}C$ approximates $\mathbb{C}L_{1}\cup\mathbb{C}L_{2}$ outside a neighborhood $U_{0}$ of this point in the sense that $\mathbb{C}C\smallsetminus U_{0}$ is a section of a tulubular neighborhood $U_{1}$ of $(\mathbb{C}L_{1}\cup\mathbb{C}L_{1})\smallsetminus U_{0}$, cf. 1.5.A. Thus $\mathbb{C}C$ may be presented as the union of two discs and a part contained in a small neighborhood of $\mathbb{C}L_{1}\cap\mathbb{C}L_{2}$. Since the whole $\mathbb{C}C$ is homeomorphic to $S^{2}$ and the complement of two disjoint discs embedded into $S^{2}$ is homeomorphic to the annulus, the third part of $\mathbb{C}C$ is an annulus. The discs are the complements of a neighborhood of $\mathbb{C}L_{1}\cap\mathbb{C}L_{2}$ in $\mathbb{C}L_{1}$ and $\mathbb{C}L_{2}$, respectively, slightly perturbed in $\mathbb{C}P^{2}$, and the annulus connects the discs through the neighborhood $U_{0}$ of $\mathbb{C}L_{1}\cap\mathbb{C}L_{2}$.

This is the complex view of the picture. Up to this point it does not matter whether the curves are defined by real equations or not.

To relate this to the real view presented in Section 1.5, one needs to describe the position of the real parts of the curves in their complexifications and the action of $conj$. It can be recovered by rough topological agruments. The whole complex picture above is invariant under $conj$. This means that the intersection point of $\mathbb{C}L_{1}$ and $\mathbb{C}L_{2}$ is real, its neighborhood $U_{0}$ can be chosen to be invariant under $conj$. Thus each half of $\mathbb{C}C$ is presented as the union of two half-discs and a half of the annulus: the half-discs approximate the halves of $\mathbb{C}L_{1}$ and $\mathbb{C}L_{2}$ and a half of annulus is contained in $U_{0}$. See Figure 15.

This is almost complete description. It misses only one point: one has to specify which half-discs are connected with each other by a half-annulus.

First, observe, that the halves of the complex point set of any curve of type I can be distinguished by the orientations of the real part. Each of the halves has the canonical orientation defined by the complex structure, and this orientation induces an orientation on the boundary of the half. This is one of the complex orientations. The other complex orientation comes from the other half. Hence the halves of the complexification are in one-to-one correspondence to the complex orientations.

Now we have an easy answer to the question above. The halves of $\mathbb{C}L_{i}$ which are connected with each other after the perturbation correspond to the complex orientations of $\mathbb{R}L_{i}$ which agree with some orientation of $\mathbb{R}C$. Indeed, the perturbed union $C$ of the lines $L_{i}$ is a curve of type I (since this is a nonempty conic, see Section 2.2). Each orientation of its real part $\mathbb{R}C$ is a complex orientation. Choose one of the orientations. It is induced by the canonical orientation of a half of the complex point set $\mathbb{C}C$. Its restriction to the part of the $\mathbb{R}C$ obtained from $\mathbb{R}L_{i}$ is induced by the orientation of the corresponding part of this half.

The union of two lines can be perturbed in two different ways. On the other hand, there are two ways to connect the halves of their complexifications. It is easy to see that different connections correspond to different perturbations. See Figure 16.

The special classical small perturbation considered above is a key for understanding what happens in the complex domain at an arbitrary classical small perturbation. First, look at the complex picture, forgetting about the real part. Take a plane projective curve, which has only nondegenerate double points. Near such a point it is organized as a union of two lines intersecting at the point. This means that there are a neighborhood $U$ of the point in $\mathbb{C}P^{2}$ and a diffeomorphism of $U$ onto $\mathbb{C}^{2}$ mapping the intersection of $U$ and the curve onto a union of two complex lines, which meet each other in 0. This follows from the complex version of the Morse lemma. By the same Morse lemma, near each double point the classical small perturbation is organized as a small perturbation of the union of two lines: the union of two transversal disks is replaced by an annulus.

For example, take the union of $m$ projective lines, no three of which have a common point. Its complex point set is the union of $m$ copies of $S^{2}$ such that any two of them have exactly one common point. A perturbation can be thought of as removal from each sphere $m-1$ disjoint discs and insertion $\frac{m(m-1)}{2}$ tubes connecting the boundary circles of the disks removed. The result is orientable (since it is a complex manifold). It is easy to realize that this is a sphere with $\frac{(m-1)(m-2)}{2}$ handles. One may prove this counting the Euler characteristic, but it may be seen directly: first, by inserting the tubes which join one of the lines with all other lines we get a sphere, then each additional tube gives rise to a handle. The number of these handles is

$\left(\begin{matrix}m-1\\ 2\end{matrix}\right)=\frac{(m-1)(m-2)}{2}.$

By the way, this description shows that the complex point set of a nonsingular plane projective curve of degree $m$ realizes the same homology class as the union of $m$ complex projective lines: the $m$-fold generator of $H_{2}(\mathbb{C}P^{2})(=\mathbb{Z})$.

Now let us try to figure out what happens with the complex schemes in an arbitrary classical small perturbation of real algebraic curves. The general case requirs some technique. Therefore we restrict ourselves to the following intermediate assertion.

Although Theorem 2.3.A describes only a very special class of classical small perturbations (namely perturbations of unions of nonsingular curves intersecting only in real points), it is enough for all constructions considered in Section 1. In Figures 17, 18, 19, 20, 21, 22 and 23 I reproduce the constructions of Figures 2, 3, 4, 6, 7, 10 and 11, enhancing them with complex orientations if the curve is of type I.

Consider an oriented topological plane curve, i. e. an oriented closed one-dimensional submanifold of the projective plane, cf. 1.2.

A pair of its ovals is said to be injective if one of the ovals is enveloped by the other.

An injective pair of ovals is said to be positive if the orientations of the ovals determined by the orientation of the entire curve are induced by an orientation of the annulus bounded by the ovals. Otherwise, the injective pair of ovals is said to be negative. See Figure 24. It is clear that the division of pairs of ovals into positive and negative pairs does not change if the orientation of the entire curve is reversed; thus, the injective pairs of ovals of a semioriented curve (and, in particular, a curve of type I) are divided into positive and negative. We let $\Pi^{+}$ denote the number of positive pairs, and $\Pi^{-}$ denote the number of negative pairs.

The ovals of an oriented curve one-sidedly embedded into $\mathbb{R}P^{2}$ can be divided into positive and negative. Namely, consider the Möbius strip which is obtained when the disk bounded by an oval is removed from $\mathbb{R}P^{2}$. If the integral homology classes which are realized in this strip by the oval and by the doubled one-sided component with the orientations determined by the orientation of the entire curve coincide, we say that the oval is negative, otherwise we say that the oval is positive. See Figure 25. In the case of a two-sided oriented curve, only the non-outer ovals can be divided into positive and negative. Namely, a non-outer oval is said to be positive if it forms a positive pair with the outer oval which envelops it; otherwise, it is said to be negative. As in the case of pairs, if the orientation of the curve is reversed, the division of ovals into positive and negative ones does not change. Let $\Lambda^{+}$ denote the number of positive ovals on a curve, and let $\Lambda^{-}$ denote the number of negative ones.

To describe a semioriented topological plane curve (up to homeomorphism of the projective plane) we need to enhance the coding system introduced in 1.2. The symbols representing positive ovals will be equipped with a superscript $+$, the symbols representing negative ovals, with a superscript $-$. This kind of code of a semioriented curve is complete in the following sense: for any two semioriented curves with the same code there exists a homeomorphism of $\mathbb{R}P^{2}$, which maps one of them to the other preserving semiorientations.

To describe the complex scheme of a curve of degree $m$ we will use, in the case of type I, the scheme of the kind described here, for its complex semiorientation, equipped with subscript I and superscript $m$ and, in the case of type II, the notation used for the real scheme, but equipped with subscript II and superscript $m$.

It is easy to check, that the coding of this kind of the complex scheme of a plane projective real algebraic curve describes the union of $\mathbb{R}P^{2}$ and the complex point set of the curve up to a homeomorphism mapping $\mathbb{R}P^{2}$ to itself.

In these notations, the complex schemes of cubic curves shown in Figure 17 are $\langle J\rangle_{II}^{3}$ and $\langle J\amalg 1^{-}\rangle_{I}^{3}$.

The complex schemes of quartic curves realized in Figure 18 are $\langle 0\rangle_{II}^{4}$, $\langle 1\rangle_{II}^{4}$, $\langle 2\rangle_{II}^{4}$, $\langle 1\langle 1^{-}\rangle\rangle_{I}^{4}$, $\langle 3\rangle_{II}^{4}$, $\langle 4\rangle_{I}^{4}$.

The complex schemes of quintic curves realized in Figure 19 are $\langle J\rangle_{II}^{5}$, $\langle J\amalg 1\rangle_{II}^{5}$, $\langle J\amalg 2\rangle_{II}^{5}$, $\langle J\amalg 1^{-}\langle 1^{-}\rangle\rangle_{I}^{5}$, $\langle J\amalg 3\rangle_{II}^{5}$, $\langle J\amalg 4\rangle_{II}^{5}$, $\langle J\amalg 1^{+}\amalg 3^{-}\rangle_{I}^{5}$ $\langle J\amalg 5\rangle_{II}^{5}$, $\langle J\amalg 3^{+}\amalg^{-}\rangle_{I}^{5}$.

In fact, these lists of complex schemes contain all schemes of nonsingular algebraic curves for degrees 3 and 5 and all nonempty schemes for degree 4. To prove this, we need not only constructions, but also restrictions on complex schemes. In the next two sections restrictions sufficient for this will be provided.

To begin with, recall the following obvious restriction, which was used in Section 2.2.

The next theorem is in a sense dual to 2.6.A.

Such a scheme exists and is unique for any $m$ (for even $m$ it is just the nest, for odd $m$ it consists of the nest and the one-sided component). To realize the scheme, perturb the union of $k$ concentric circles and, in the case of odd $m$, a line disjoint from the circles. The uniqueness was proved in 1.3, see 1.3.C.

I preface the proof of 2.6.D with a construction interesting for its own. It provides a kind of window through which one can take a look at the imaginary part of $\mathbb{C}P^{2}$.

As we know (see Section 2.2), the complex point set of a real line is divided by its real point set into two halves, which are in a natural one-to-one correspondence with the orientations of the real line. The set of all real lines on the projective plane is the real point set of the dual projective plane. The halves of lines comprise a two-dimensional sphere covering this projective plane. An especially clear picture of these identifications appears, if one identifies real lines on the projective plane with real planes in $\mathbb{R}^{3}$ containing 0. A half of a line is interpreted as the corresponding plane with orientation. An oriented plane corresponds to its positive unit normal vector, which is nothing but a point of $S^{2}$. The complex conjugation $conj$ maps a half of a real line to the other half of the same line. It corresponds to the reversing of the orientation, which, in turn, corresponds to the antipodal involution $S^{2}\to S^{2}:x\mapsto-x$.

There is a unique real line passing through any imaginary point of $\mathbb{C}P^{2}$. To construct such a line, connect the point with the conjugate one. The connecting line is unique since a pair of distinct points determines a line, and this line is real, since it coincides with its image under $conj$.

Consequently, there is a unique half of a real line containing an imaginary point of $\mathbb{C}P^{2}$. This construction determines a fibration $p:\mathbb{C}P^{2}\smallsetminus\mathbb{R}P^{2}\to S^{2}$. The fibres of $p$ are the halves of real lines. Note that conjugate points of $\mathbb{C}P^{2}\smallsetminus\mathbb{R}P^{2}$ are mapped to antipodal points of $S^{2}$.

Now we shall consider a powerful restriction on a complex orientation of a curve of type I. It is powerful enough to imply restrictions even on real schemes of type I. The first version of this restriction was published in 1974, see Rok-74. There Rokhlin considered only the case of an algebraic M-curve of even degree. In Mis-75 Mishachev considered the case of an algebraic M-curve of odd degree. For an arbitrary nonsingular algebraic curve of type I, it was formulated by Rokhlin Rok-78 in 1978. The proofs from Rok-74 and Mis-75 work in this general case. The only reason to restrict the main formulations in these early papers to M-curves was the traditional viewpoint on the subject of the topology of real plane algebraic curves.

Here are Rokhlin’s formulations from Rok-78.

Theorems 2.7.A and 2.7.B can be united into a single formulation. This requires, however, two preliminary definitions.

First, given an oriented topological curve $C$ on $\mathbb{R}P^{2}$, for any point $x$ of its complement, there is the index $i_{C}(x)$ of the point with respect to the curve. It is a nonnegative integer defined as follows. Draw a line $L$ on $\mathbb{R}P^{2}$ through $x$ transversal to $C$. Equip it with a normal vector field vanishing only at $x$. For such a vector field, one may take the velocity field of a rotation of the line around $x$. At each intersection point of $L$ and $C$ there are two directions transversal to $L$: the direction of the vector belonging to the normal vector field and the direction defined by the local orientation of $C$ at the point. Denote the number of intersection points where the directions are faced to the same side of $L$ by $i_{+}$ and the number of intersection points where the directions are faced to the opposite sides of $L$ by $i_{-}$. Then put $i_{C}(x)=|i_{+}-i_{-}|/2$.⁵⁵Division by 2 appears here to make this notion closer to the well-known notion for an affine plane curve. In the definition for affine situation one uses a ray instead of entire line. In the projective situation there is no natural way to divide a line into two rays, but we still have an opportunity to divide the result by 2. Another distinction from the affine situation is that there the index may be negative. It is related to the fact that the affine plane is orientable, while the projective plane is not. It is easy to check that $i_{C}(x)$ is well defined: it depends neither on the choice of $L$, nor on the choice of the normal vector field. It does not change under reversing of the orientation of $C$. Thus for any nonsingular curve $A$ of type I on the complement $\mathbb{R}P^{2}\smallsetminus\mathbb{R}A$, one has well defined function $i_{\mathbb{R}A}$.

The second prerequisite notion is a sort of unusual integration: an integration with respect to the Euler characteristic, in which the Euler characteristic plays the role of a measure. It is well known that the Euler characteristic shares an important property of measures: it is additive in the sense that for any sets $A$, $B$ such that the Euler characteristics $\chi A$, $\chi B$, $\chi(A\cap B)$ and $\chi(A\cup B)$ are defined,

$\chi(A\cup B)=\chi(A)+\chi(B)-\chi(A\cap B).$

However, the Euler characteristic is neither $\sigma$-additive, nor positive. Thus the usual theory of integral cannot be applied to it. This can be done though if one restricts to a very narrow class of functions. Namely, to functions which are finite linear combinations of characteristic functions of sets belonging to some algebra of subsets of a topological space such that each element of the algebra has a well defined Euler characteristic. For a function $f=\sum_{i=1}^{r}\lambda_{i}\hbox{\rm\hbox to 0.0pt{1} {{\rm I}}}_{S_{i}}$ set

$\int f(x)\,d\chi(x)=\sum_{i=1}^{r}\lambda_{i}\chi(S_{i}).$

For details and applications of that notion, see Vir-88.

Now we can unite 2.7.A and 2.7.B:

Here I give a proof of 2.7.C, skipping the most complicated details. Take a curve $A$ of degree $m$ and type I. Let $\mathbb{C}A_{+}$ be its half bounded by $\mathbb{R}A$. It may be considered as a chain with integral coefficients. The boundary of this chain (which is $\mathbb{R}A$ equipped with the complex orientation) bounds in $\mathbb{R}P^{2}$ a chain $c$ with rational coefficients, since $H_{1}(\mathbb{R}P^{2};\,\mathbb{Q})=0$. In fact, in the case of even degree the chain can be taken with integral coefficients, but in the case of odd degree the coefficients are necessarily half-integers. The explicit form of $c$ may be given in terms of function $i_{\mathbb{R}A}$: it is a linear combination of the fundamental cycles of the components of $\mathbb{R}P^{2}\smallsetminus\mathbb{R}A$ with coefficients equal to the values of $i_{\mathbb{R}A}$ on the components (taken with appropriate orientations).

Now take the cycle $[\mathbb{C}A_{+}]-c$ and its image under $conj$, and calculate their intersection number in two ways.

First, it is easy to see that the homology class $\xi$ of $[\mathbb{C}A_{+}]-c$ is equal to

$$\frac{1}{2}[\mathbb{C}A]=\frac{m}{2}[\mathbb{C}P^{1}]\in H_{2}(\mathbb{C}P^{2} ;\,\mathbb{Q})$$ Indeed, $$[\mathbb{C}A_{+}]-c-conj([\mathbb{C}A_{+}]-c)=[\mathbb{C}A]+c-conj(c)=[\mathbb {C}A]$$ , and therefore $$\xi-conj_{*}(\xi)=[\mathbb{C}A]=m[\mathbb{C}P^{1}]\in H_{2}(\mathbb{C}P^{2}).$$ On the other hand, $conj$ acts in $H_{2}(\mathbb{C}P^{2})$ as multiplication by $-1$, and hence $\xi-conj_{*}(\xi)=2\xi=m[\mathbb{C}P^{1}]$. Therefore $\xi\circ conj_{*}(\xi)=-(\frac{m}{2})^{2}$.

Second, one may calculate the same intersection number geometrically: moving the cycles into a general position and counting the local intersection numbers. I will perturb the cycle $[\mathbb{C}A_{+}]-c$. First, choose a smooth tangent vector field $V$ on $\mathbb{R}P^{2}$ such that it has only nondegenerate singular points, the singular points are outside $\mathbb{R}A$, and on $\mathbb{R}A$ the field is tangent to $\mathbb{R}A$ and directed according to the complex orientation of $A$ which comes from $\mathbb{C}A_{+}$. The latter means that at any point $x\in\mathbb{R}A$ the vector $\sqrt{-1}V(x)$ is directed inside $\mathbb{C}A_{+}$ (the multiplication by $\sqrt{-1}$ makes a real vector normal to the real plane and lieves any vector tangent to $\mathbb{R}A$ tangent to $\mathbb{C}A$). Now shift $\mathbb{R}A$ inside $\mathbb{C}A_{+}$ along $\sqrt{-1}V$ and extend this shift to a shift of the whole chain $c$ along $\sqrt{-1}V$. Let $c^{\prime}$ denote the result of the shift of $c$ and $h$ denote the part of $\mathbb{C}A_{+}$ which was not swept during the shift. The cycle $[h]-c^{\prime}$ represents the same homology class $\xi$ as $[\mathbb{C}A_{+}]-c$, and we can use it to calculate the intersection number $\xi\circ conj_{*}(\xi)$. The cycles $[h]-c^{\prime}$ and $conj([\mathbb{C}A_{+}]-c)$ intersect only at singular points of $V$. At a singular point $x$ they are smooth transversal two-dimensional submanifolds, each taken with multiplicity $-i_{\mathbb{R}A}(x)$. The local intersection number at $x$ is equal to $(i_{\mathbb{R}A}(x))^{2}$ multiplied by the local intersection number of the submanifolds supporting the cycles. The latter is equal to the index of the vector field $V$ at $x$ multiplied by $-1$.

I omit the proof of the latter statement. It is nothing but a straightforward checking that multiplication by $\sqrt{-1}$ induces isomorphism between tangent and normal fibrations of $\mathbb{R}A$ in $\mathbb{C}A$ reversing orientation.

Now recall that the sum of indices of a vector field tangent to the boundary of a compact manifold is equal to the Euler characteristic of the manifold. Therefore the input of singular points lying in a connected component of $\mathbb{R}P^{2}\smallsetminus\mathbb{R}A$ is equal to the Euler characteristic of the component multiplied by $-(i_{\mathbb{R}A}(x))^{2}$ for any point $x$ of the component. Summation over all connected components of $\mathbb{R}P^{2}\smallsetminus\mathbb{R}A$ gives $-\int(i_{\mathbb{R}A}(x))^{2}\,d\chi(x)$. Its equality to the result of the first calculation is the statement of 2.7.C.$\square$

Denote the number of all injective pairs of ovals for a curve under consideration by $\Pi$.

As it was promised in Section 2.5, we can prove now that only schemes realized in Figures 17, 18 and 19 are realizable by curves of degree 3, 4 and 5, respectively. For reader’s convinience, I present here a list of all these complex schemes in Table 5.

Degree 3. By Harnack’s inequality, the number of components is at most 2. By 1.3.B a curve of degree 3 is one-sided, thereby the number of components is at least 1. In the case of 1 component the real scheme is $\langle J\rangle$, and the type is II by Klein’s congruence 2.6.C. In the case of 2 components the type is I by 2.6.B. The real scheme is $\langle J\amalg 1\rangle$. Thus we have 2 possible complex schemes: $\langle J\amalg 1^{-}\rangle_{I}^{3}$ (realized above) and $\langle J\amalg 1^{+}\rangle_{I}^{3}$. For the first one $\int(i_{\mathbb{R}A}(x))^{2}\,d\chi(x)=9/4$ and for the second $\int(i_{\mathbb{R}A}(x))^{2}\,d\chi(x)=1/4$. Since the right hand side of the complex orientation formula is $m^{2}/4$ and $m=3$, only the first possibility is realizable.$\square$

Degree 4. By Harnack’s inequality the number of components is at most 4. We know (see 1.4) that only real schemes $\langle 0\rangle$, $\langle 1\rangle$, $\langle 2\rangle$, $\langle 1\langle 1\rangle\rangle$, $\langle 3\rangle$ and $\langle 4\rangle$ are realized by nonsingular algebraic curves of degree 4. From Klein’s congruence 2.6.C it follows that the schemes $\langle 1\rangle$ and $\langle 3\rangle$ are of type II. The scheme $\langle 0\rangle$ is of type II by 2.6.A. By 2.6.B $\langle 4\rangle$ is of type I.

The scheme $\langle 2\rangle$ is of type II, since it admits no orientation satisfying the complex orientation formula. In fact, for any orientation $\int(i_{\mathbb{R}A}(x))^{2}\,d\chi(x)=2$ while the right hand side is $m^{2}/4=4$.

By 2.6.D the scheme $\langle 1\langle 1\rangle\rangle$ is of type I. A calculation similar to the calculation above on the scheme $\langle 2\rangle$, shows that only one of the two semiorientations of the scheme $\langle 1\langle 1\rangle\rangle$ satisfies the complex orientation formula. Namely, $\langle 1\langle 1^{-}\rangle\rangle$. It was realized in Figure 18.

Degree 5. By Harnack’s inequality the number of components is at most 7. We know (see 1.4) that only real schemes $\langle J\rangle$, $\langle J\amalg 1\rangle$, $\langle J\amalg 2\rangle$, $\langle J\amalg 1\langle 1\rangle\rangle$, $\langle J\amalg 3\rangle$, $\langle J\amalg 4\rangle$, $\langle J\amalg 5\rangle$, $\langle J\amalg 6\rangle$ are realized by nonsingular algebraic curves of degree 5. From Klein’s congruence 2.6.C it follows that the schemes $\langle J\amalg 1\rangle$, $\langle J\amalg 3\rangle$, $\langle J\amalg 5\rangle$ are of type II. By 2.7.F $\langle J\rangle$ and $\langle J\amalg 2\rangle$ are of type II.

By 2.6.B $\langle J\amalg 6\rangle$ is of type I. The complex orientation formula gives the value of $\Lambda_{-}$ (cf. Proof of 2.7.F): $\Lambda_{-}=\frac{1}{2}k(k+1)=3$. This determines the complex scheme. It is $\langle J\amalg 3^{-}\amalg 3^{+}\rangle_{I}^{5}$.

By 2.6.D $\langle J\amalg 1\langle 1\rangle\rangle$ is of type I. The complex orientation formula allows only the semiorientation with $\Lambda_{-}=2$. Cf. Figure 19.

The real scheme $\langle J\amalg 4\rangle$ is of indefinite type, as follows from the construction shown in Figure 19. In the case of type I only one semiorientation is allowed by the the complex orientation formula. It is $\langle J\amalg 3^{-}\amalg 1^{+}\rangle_{I}^{5}$.

2.1 Prove that for any two semioriented curves with the same code (of the kind introduced in 3.7) there exists a homeomorphism of $\mathbb{R}P^{2}$ which maps one of them to another preserving semiorientations.

2.2  Prove that for any two curves $A_{1}$, $A_{2}$ with the same code of their complex schemes (see Subsection 2.5) there exists a homeomorphism $\mathbb{C}A_{1}\cup\mathbb{R}P^{2}\to\mathbb{C}A_{2}\cup\mathbb{R}P^{2}$ commuting with $conj$.

2.3  Deduce 2.7.A and 2.7.B from 2.7.C and, vise versa, 2.7.C from 2.7.A and 2.7.B.

In Section 1 all prohibitions were deduced from the Bézout Theorem. In Section 2 many proofs were purely topological. A straightforward analysis shows that the proofs of all prohibitions are based on a small number of basic properties of the complexification of a nonsingular plane projective algebraic curve. It is not difficult to list all these properties of such a curve $A$:

1. (1)

   Bézout’s theorem;

2. (2)

   $\mathbb{C}A$ realizes the class $m[\mathbb{C}P^{1}]\in H_{2}(\mathbb{C}P^{2})$;

3. (3)

   $\mathbb{C}A$ is homeomorphic to a sphere with $(m-1)(m-2)/2$ handles;

4. (4)

   $conj(\mathbb{C}A)=conj$;

5. (5)

   the tangent plane to $\mathbb{C}A$ at a point $x\in\mathbb{R}A$ is the complexification of the tangent line of $\mathbb{R}A$ at $x$.

The last four are rough topological properties. Bézout’s theorem occupies a special position. If we assume that some surface smoothly embedded into $\mathbb{C}P^{2}$ intersects the complex point set of any algebraic curve as, according to Bézout’s theorem, the complex point set of an algebraic curve, then this surface is the complex point set of an algebraic curve. Thus the Bézout theorem is completely responsible for the whole set of properties of algebraic curves. On the other hand, its usage in obtaining prohibitions involves a construction of auxiliary curves, which may be very subtle.

Therefore, along with algebraic curves, it is useful to consider objects which imitate them topologically.

An oriented smooth closed connected two-dimensional submanifold $S$ of the complex projective plane $\mathbb{C}P^{2}$ is called a flexible curve of degree $m$ if:

1. (i)ii

   $S$ realizes $m[\mathbb{C}P^{1}]\in H_{2}(\mathbb{C}P^{2})$;

2. (ii)i

   the genus of $S$ is equal to $(m-1)(m-2)/2$;

3. (iii)

   $conj(S)=S$;

4. (iv)

   the field of planes tangent to $S$ on $S\cap\mathbb{R}P^{2}$ can be deformed in the class of planes invariant under $conj$ into the field of (complex) lines in $\mathbb{C}P^{2}$ which are tangent to $S\cap\mathbb{R}P^{2}$.

A flexible curve $S$ intersects $\mathbb{R}P^{2}$ in a smooth one-dimensional submanifold, which is called the real part of $S$ and denoted by $\mathbb{R}S$. Obviously, the set of complex points of a nonsingular algebraic curve of degree $m$ is a flexible curve of degree $m$. Everything said in Section 2.1 about algebraic curves and their (real and complex) schemes carries over without any changes to the case of flexible curves. We say that a prohibition on the schemes of curves of degree $m$ comes from topology if it can be proved for the schemes of flexible curves of degree $m$. The known classification of schemes of degree $\leq 6$ can be obtained using only the prohibitions that come from topology. In other words, for $m\leq 6$ all prohibitions come from topology.

The simplest prohibitions are not related to the position of $\mathbb{R}S$ in $\mathbb{R}P^{2}$, but deal with the following purely topological situation: a surface $S$, which is homeomorphic to a sphere with $g$ ($=(m-1)(m-2)/2$) handles, and an involution $c$ ($=conj$) of $S$ reversing orientation with fixed point set $F$ ($=\mathbb{R}S$).

The most important of these prohibitions is Harnack’s inequality. Recall that it is

$L\leq\frac{(m-1)(m-2)}{2}+1,$

where $L$ is the number of connected components of the real part a curve and $m$ is its degree. Certainly, this formulation given in Section 1.3 can be better adapted to the context of flexible curves. The number $\frac{(m-1)(m-2)}{2}$ is nothing but the genus. Therefore the Harnack inequality follows from the following theorem.

In turn, 3.2.A can be deduced from the following purely topological theorem on involutions:

This theorem is one of the most famous results of the Smith theory. It is deduced from the basic facts on equivariant homology of involution, see, e. g., Bre-72, Chapter 3.

Theorem 3.2.A follows from 3.2.B, since

$\dim_{\mathbb{Z}_{2}}H_{*}(S;\,\mathbb{Z}_{2})=2+2g,$

and

$\dim_{\mathbb{Z}_{2}}H_{*}(F;\,\mathbb{Z}_{2})=2L.$

Smith - Floyd Theorem can be applied to high-dimensional situation, too. See Sections 5.3 and LABEL:s15.1d. In the one-dimensional case, which we deal with here, Theorem 3.2.B is easy to prove without any homology tool, like the Smith theory. Namely, consider the orbit space $S/c$ of the involution. It is a connected surface with boundary. The boundary is the image of the fixed point set. The Euler characteristic of the orbit space is equal to the half of the Euler characteristic of $S$, i.e. it is $\frac{2-2g}{2}=1-g$. Cap each boundary circle with a disk. The result is a closed connected surface with Euler characteristic $1-g+L$. On the other hand, as it is well known, the Euler characteristic of a connected closed surface is at most 2. (Remind that such a surface is homeomorphic either to the sphere, which has Euler characteristic 2, or the sphere with $h$ handles, whose Euler characteristic is $2-2h$, or sphere with $h$ Möbius strips having Euler characteristic $2-h$.) Therefore $1-g+L\leq 2$, and $L\leq g+1$.$\square$

These arguments contain more than just a proof of 2.3.A. In particular, they imply that

Similarly, in the case of an $(M-1)$-curve, the orbit space is homeomorphic to the projective plane with holes.

If $F$ separates $S$ (i.e., $S\smallsetminus F$ is not connected), the involution $c$ is said to be of type I, otherwise it is said to be of type II. The types correspond to the types of real algebraic curves (see Section 2.1).

Note that $F$ separates $S$ at most into two pieces. To prove this, we can use the same arguments as in a footnote in Section 2.1: the closure of tne union of a connected component of $S\smallsetminus F$ with its image under $c$ is open and close in $S$, but $S$ is connected.

Now consider the simplest prohibition involving the placement of the real part of the flexible curve in the projective plane.

Rokhlin’s complex orientation formula also comes from topology. The proof presented in Section 2.7 works for a flexible curve.

At this point I want to break a textbook style exposition. Escaping a detailed exposition of prohibitions, I switch to a survey.

In the next two sections, the current state of prohibitions on the topology of a flexible curve of a given degree is outlined. (Recall that all formulations of this sort are automatically valid for real projective algebraic plane curves of the same degree.) After the survey a light outline of some proofs is proposed. It is included just to convey a general impression, rather than for more serious purposes. For complete proofs, see the surveys Wil-78, Rok-78, Arn-79, Kha-78, Kha-86, Vir-86 and the papers cited there.

In this section I list all prohibitions on the real scheme of a flexible curve of degree $m$ that I am aware of, including the ones already referred to above, but excluding prohibitions which follow from the other prohibitions given here or from the prohibitions on the complex schemes which are given in the next section.

This fact was given before as a corollary of Bézout’s theorem (see Section 1.3) and proved for flexible curves in Section 3.2 (Theorem 3.2.G).

Harnack’s inequality is undoubtedly the best known and most important prohibition. It can also be deduced from Bézout’s theorem (cf. Section 1.3) and was proved for flexible curves in Section 3.2 (Theorem 3.2.A).

In prohibitions 3.3.C–3.3.P the degree $m$ of the curve is even: $m=2k$.

Extremal Properties of Harnack’s Inequality

The Euler characteristic of a component of the complement of a curve in $\mathbb{R}P^{2}$ is called the characteristic of the oval which bounds the component from outside. An oval with a positive characteristic is said to be elliptic, an oval with the zero characteristic is said to be parabolic and an oval with a negative characteristic is said to be hyperbolic.

Denote the number of even ovals with positive characteristic by $p^{+}$, the number of even ovals with zero characteristic by $p^{0}$, and the number of even ovals with negative characteristic by $p^{-}$. Similarly define $n^{+},n^{0}$ and $n^{-}$ for the odd ovals; and let $l^{+},l^{0}$ and $l^{-}$ be the corresponding numbers for both even and odd ovals together.

Refined Petrovsky Inequalities

Refined Arnold Inequalities

Extremal Properties of the Refined Arnold Inequalities

Viro-Zvonilov Inequalities

Besides Harnack’s inequality, we know only one family of prohibition coming from topology which extends to real schemes of both even and odd degree. For proofs see VZ-92.

The latter inequality also holds true for even $m\neq 4$, but it follows from Arnold inequalities 3.3.J and 3.3.K.

If $m$ is even, this inequality follows from 3.3.J–3.3.L.

Recall that $l$ denotes the total number of ovals on the curve. The following theorem is a reformulation of 3.2.E.

Extremal Properties of Harnack’s Inequality

Extremal Properties of the Refined Arnold Inequalities

Extremal Properties of the Viro-Zvonilov Inequality

Congruences

The next two congruences are included violating a general promise given at the beginning of the previous section. There I promised exclude prohibitions which follow from other prohibitions given here. The following two congruences are consequences of Rokhlin’s formula 3.4.C. The first of them was discovered long before 3.4.C. The second was overlooked by Rokhlin in Rok-74, where he even mistakenly proved that such a result cannot exist. Namely, Rokhlin proved that the complex orientation formula does not imply any result which would not follow from the prohibitions known by that time and could be formulated solely in terms of the real scheme. Slepian congruence 3.4.K for M-curves is the only counter-example to this Rokhlin’s statement. Slepian was Rokhlin’s student, he discovered a gap in Rokhlin’s arguments and deduced 3.4.K.

Rokhlin Inequalities

Denote by $\pi$ and $\nu$ the number of even and odd nonempty ovals, respectively, bounding from the outside those components of the complement of the curve which have the property that each of the ovals bounding them from the inside envelops an odd number of other ovals.

Theorems formulated in 3.3 and 3.4 are very different in their profundity. The simplest of them were considered in Subsection 3.2.

Congruences

There are two different approaches to proving congruences. The first is due basically to Arnold Arn-71 and Rokhlin Rok-72. It is based on consideration of the intersection form of two-fold covering $Y$ of $\mathbb{C}P^{2}$ branched over the complex point set of the curve. The complex conjugation involution $conj:\mathbb{C}P^{2}\to\mathbb{C}P^{2}$ is lifted to $Y$ in two different ways, and the liftings induce involutions in $H_{2}(Y)$, which are isometries of the intersection form. One has to take an appropriate eigenspace of one of the liftings and consider the restriction of the intersection form to the eigenspace. The signature of this restriction can be calculated in terms of $p-n$. On the other hand, it is involved into some congruences of purely arithmetic nature relating it with the discriminant of the form and the value of the form on some of characteristic vectors. The latters can be calculated sometimes in terms of degree and the difference between the number of ovals and the genus of curve. Realizations of this scheme can be found in Arn-71 for 3.4.J, Rok-72 for 3.3.C, Kha-73 and GK-73 for 3.3.D, Nik-83 for 3.3.F, 3.3.G, 3.4.I and a weakened form of 3.3.E. In survey Wil-78 this method was used for proving 3.3.C, 3.3.D and 3.4.J.

The second approach is due to Marin Mar-80. It is based on application of the Rokhlin-Guillou-Marin congruence modulo 16 on characteristic surface in a 4-manifold, see GM-77. It is applied either to the surface in the quotient space $\mathbb{C}P^{2}/conj$ (diffeomorphic to $S^{4}$) made of the image of the flexible curve $S$ and a half of $\mathbb{R}P^{2}$ bounded by $\mathbb{R}S$ (as it is the case for proofs of 3.3.C, 3.3.D and 3.4.E in Mar-80), or to the surface in $\mathbb{C}P^{2}$ made of a half of $S$ and a half of $\mathbb{R}P^{2}$ (as it is the case for proofs of 3.3.E, 3.4.I and special cases of 3.3.F and 3.3.G in Fie-83).

The first approach was applied also in high-dimensional situations. The second approach worked better than the second one for curves on surfaces distinct from projective plane, see Mik-94. Both were used for singular curves KV-88.

Inequalities

Inequalities 3.3.H, 3.3.I, 3.3.J, 3.3.K, 3.4.J and 3.4.K are proved along the same scheme, originated by Arnold Arn-71. One constructs an auxiliary manifold, which is the two-fold covering of $\mathbb{C}P^{2}$ branched over $S$ in the case of 3.3.H, 3.3.I, 3.3.J and 3.3.K and the two-fold covering of $\mathbb{C}P^{2}/conj$ branched over the union of $S/conj$ and a half of $\mathbb{R}P^{2}$ in the case of 3.4.J and 3.4.K. Then preimages of some of the components of $\mathbb{R}P^{2}\smallsetminus\mathbb{R}S$ gives rise to cycles in this manifold. Those cycles define homology classes with special properties formulated in terms of their behavior with respect to the intersection form and the complex conjugation involutions. On the other hand, the numbers of homology classes with these properties are estimated. See Arn-71, Gud-74, Wil-78 and Rok-80.

In this subsection, I show that for degrees $\leq 5$ the prohibitions coming from topology allow the same set of complex schemes as all prohibitions. The set of complex schemes of algebraic curves of degrees $\leq 5$ was described in 2.8. In fact the same is true for degree 6 too. For degree greater than 6, it is not known, but there is no reason to believe that it is the case.

Degrees $\leq 3$. Theorems 3.3.A and the Harnack inequality 3.3.B prohibit all non realizable real schemes for degree $\leq 3$. To obtain the complete set of prohibitions for complex schemes of degrees $\leq 3$ one has to add the Klein congruence 3.4.B, 3.4.D and the complex orientation formula 3.4.C; cf. Section 2.8.

Degree 4. By the Arnold inequlity 3.3.K, a flexible curve of degree 4 cannot have a nest of depth 3. By the Arnold inequality 3.3.J, it has at most one nonempty positive oval, and if it has a nonempty oval then, by the extremal property 3.3.L of this inequality, the real scheme is $\langle 1\langle 1\rangle\rangle$. Together with 3.3.A and the Harnack inequality 3.3.B, this forms the complete set of prohibitions for real schemes of degree 4.

From the Klein congruence 3.4.B, it follows that the real schemes $\langle 1\rangle$ and $\langle 3\rangle$ are of type II. The empty real scheme $\langle 0\rangle$ is of type II by 3.4.A. By the extremal property 3.4.D of the Harnack inequality, $\langle 4\rangle$ is of type I. The real scheme $\langle 2\rangle$ is of type II by the complex orientation formula 3.4.C, cf. Section 2.8. By 3.4.F, the scheme $\langle 1\langle 1\rangle\rangle$ is of type I. By the complex orientation formula, it admits only the complex orientation $\langle 1\langle 1^{-}\rangle\rangle$.

Degree 5. By the Viro-Zvonilov inequality 3.3.O, a flexible curve of degree 5 can have at most one nonempty oval. By the extremal property of this inequality 3.3.P, if a flexible curve of degree 5 has a nonempty oval, then its real scheme is $\langle J\amalg 1\langle 1\rangle\rangle$. Together with 3.3.A and the Harnack inequality 3.3.B, this forms the complete set of prohibitions for real schemes of degree 5.

From the Klein congruence 3.4.B, it follows that the real schemes $\langle J\amalg 1\rangle$, $\langle J\amalg 3\rangle$, and $\langle J\amalg 5\rangle$ are of type II. From the complex orientation formula, one can deduce that the real schemes $\langle J\rangle$ and $\langle J\amalg 2\rangle$ are of type II, cf. 2.8. By the extremal property 3.4.D of the Harnack inequality, $\langle J\amalg 6\rangle$ is of type I. The complex orientation formula allows only one complex semiorientation for this scheme, namely $\langle J\amalg 3^{-}\amalg 3^{+}\rangle$. By the 3.4.H, the real scheme $\langle J\amalg 1\langle 1\rangle\rangle$ is of type I. The complex orientation formula allows only one complex semiorientation for this scheme, namely $\langle J\amalg 1^{-}\langle 1^{-}\rangle\rangle$, cf. 2.8. The real scheme $\langle J\amalg 4\rangle$ is of indefinite type (even for algebraic curves, see 2.8). In the case of type I, only one semiorientation is allowed by the the complex orientation formula. It is $\langle J\amalg 3^{-}\amalg 1^{+}\rangle$.

The arsenal of constructions in Section 1 and the supply of curves constructed there, which are very modest from the point of view of classification problems, turn out to be quite rich if we are interested in the problem of sharpness of the inequalities in Section 3.3.

The Harnack curves of even degree $m$ with scheme

$\langle(3m^{2}-6m)/8\amalg 1\langle m^{2}-6m+8)/8\rangle\rangle$

which were constructed in Section 1.6 (see also Section 1.9) not only show that Harnack’s inequality 3.3.B is the best possible, but also show the same for the refined Petrovsky inequality 3.3.H.

One of the simplest variants of Hilbert’s construction (see Section 1.10) leads to the construction of a series of M-curves of degree $m\equiv 2\mod 4$ with scheme $\left\langle\frac{(m-2)(m-4)}{8}\amalg 1\left\langle\frac{3m(m-2)}{8}\right \rangle\right\rangle$. This proves that the refined Petrovsky inequality 3.3.I for $m\equiv 2\mod 4$ is sharp. If $m\equiv 0\mod 4$, the methods of Section 1 do not show that this inequality is the best possible. This fact will be proved below in LABEL:s5.4s.

The refined Arnold inequality 3.3.J is best possible for any even $m$. If $m\equiv 2\mod 4$, this can be proved using the Wiman M-curves (see the end of Section 1.12). If $m\equiv 0\mod 4$, it follows using curves obtained from a modification of Wiman’s construction: the construction proceeds in exactly the same way, except that the opposite perturbation is taken, as a result of which one obtains a curve that can serve as the boundary of a tubular neighborhood of an M-curve of degree $m/2$.

The last construction (doubling), if applied to an M-curve of odd degree, shows that the refined Arnold inequality 3.3.K is the best possible for $m\equiv 2\mod 4$. If $m\equiv 0\mod 4$, almost nothing is known about sharpness of the inequality 3.3.K, except that for $m=8$ the right side can be lowered by 2.

In conclusion of this section, let us come back to algebraic curves. We see that to a great extent the topology of their real point sets is determined by the properties which were included into the definition of flexible curves. In fact, it has not been proved that it is not determined by these properties completely. However some known prohibitions on topology of real algebraic curves have not been deduced from them.

As a rule, these prohibitions are hard to summarize, in the sense that it is difficult to state in full generality the results obtained by some particular method. To one extent or another, all of them are consequences of Bézout’s theorem.

Consider first the restrictions which follow directly from the Bézout theorem. To state them, we introduce the following notations. Denote by $h_{r}$ the maximum number of ovals occurring in a union of $\leq r$ nestings. Denote by $h^{\prime}_{r}$ the maximum number of ovals in a set of ovals contained in a union of $\leq r$ nests but not containing an oval which envelops all of the other ovals in the set. Under this notations Theorems 1.3.C and 1.3.D can be stated as follows:

These statements suggest a whole series of similar assertions. Denote by $c(q)$ the greatest number $c$ such that there is a connected curve of degree $q$ passing through any $c$ points of $\mathbb{R}P^{2}$ in general position. It is known that $c(1)=2$, $c(2)=5$, $c(3)=8$, $c(4)=13$

The following two restrictions on complex schemes are similar to Theorems 3.8.A and 3.8.B. However, I do not know the corresponding analogues of 3.8.C and 3.8.D.

Here I will not even try to discuss the most general prohibitions which do not come from topology. I will only give some statements of results which have been obtained for curves of small degree.

Proofs of 3.8.G and 3.8.H are based on technique initiated by Fiedler Fie-82. It will be developed in the next Section.

To begin with, I present another proof of Theorem 2.6.D. It gives slightly more: not only that a curve with maximal nest has type I, but that its complex orientation is unique. This is not difficult to obtain from the complex orientation formula. The real cause for including this proof is that it is the simplest application of the technique, which will work in this section in more complicated situations. Another reason: I like it.

Recall that by Corollary 1.3.C of the Bézout theorem a nest of a curve of degree $m$ has depth at most $m/2$, and if a curve of degree $m$ has a nest of depth $[m/2]$, then it does not have any ovals not in the nest. Thus the real scheme of a curve of 4.1.A is $\langle 1\langle 1\dots\langle 1\rangle\dots\rangle\rangle$, if $m$ is even, and $\langle J\langle 1\langle\dots 1\langle 1\rangle\dots\rangle\rangle$ if $m$ is odd. Theorem 4.1.A says that the complex scheme in this case is defined by the real one and it is

$\langle 1\langle 1^{-}\dots\langle 1^{-}\rangle\dots\rangle\rangle_{I}^{m}$

for even $m$ and

$\langle J\langle 1^{-}\langle\dots 1^{-}\langle 1^{-}\rangle\dots\rangle \rangle_{I}^{m}$

if $m$ is odd.

Consider the pencil of real lines passing through the intersection point of real lines $L_{0}$, $L_{1}$. It is divided by $L_{0}$ and $L_{1}$ into two segments. Each of the segments can be described as $\{L_{t}\}_{t\in[0,1]}$, where $L_{t}$ is defined by equation $(1-t)\lambda_{0}(x)+t\lambda_{1}(x)=0\}$ under an appropriate choice of equations $\lambda_{0}(x)=0$ and $\lambda_{1}(x)=0$ defining $L_{0}$ and $L_{1}$, respectively. Such a family $\{L_{t}\}_{t\in[0,1]}$ is called a segment of the line pencil connecting $L_{0}$ with $L_{1}$.

A point of tangency of two oriented curves is said to be positive if the orientations of the curves define the same orientation of the common tangent line at the point, and negative otherwise.

The following theorem is a special case of the main theorem of Fiedler’s paper Fie-82.

I give here a proof, which is less general than Fiedler’s original one. I hope though that it is more visualizable.

Roughly speaking, the main idea of this proof is that, looking at a curve, it is useful to move slightly the viewpoint. When one looks at the intersection of the complexification of a real curve with complexification of real lines of some pencil, besides the real part of the curve only some arcs are observable. These arcs connect ovals of the curve, but they do not allow to realize behavior of the complexification around. However, when the veiwpoint (= the center of the pencil) is moving, the arcs are moving too sweeping ribbons in the complexification. The ribbons bear orientation inherited from the complexification and thereby they allow to trace relation between the induced orientation of the ovals connected by the arcs. See Figure 26

The next thing to do is to obtain prohibitions on complex schemes using Fiedler’s theorem. It takes some efforts because we want to deduce topological results from a geometric theorem. In the theorem it is crucial how the curve is positioned with respect to lines, while in any theorem on topology of a real algebraic curve, the hypothesis can imply some particular position with respect to lines only implicitely.

Let $A$ be a nonsingular curve of type I and $P\in\mathbb{R}P^{2}\smallsetminus\mathbb{R}A$. Let $Z=\{L_{t}\}_{t\in I}$ be a segment of the pencil of lines passing through $P$, which contains neither a line tangent to $\mathbb{R}A$ at a point of inflection of $\mathbb{R}A$ nor a line, whose complexifications is tangent to $\mathbb{C}A$ at an imaginary point. Denote $\bigcup_{t\in I}\mathbb{R}L_{t}$ by $C$.

Fix a complex orientations of $A$ and orientations of the lines $\mathbb{R}L_{t}$, $t\in I$, which turn to one another under the natural isotopy. Orient the part $C$ of the projective plane in such a way that this orientation induces on $\mathbb{R}L_{0}$, as on a part of its boundary, the orientation selected above. An oval of $A$, lying in $C$ is said to be positive with respect to $Z$ if its complex orientation and orientation of $C$ induce the same orientation of its interior; otherwise the oval is said to be negative with respect to $Z$.

A point of tangency of $A$ and a line from $Z$ is a nondegenerate critical point of the function $A\cap C\to I$ which assigns to $x$ the real number $t\in I$ such that $x\in L_{t}$. By index of the point of tangency we shall call the Morse index of this function at that point (zero, if it is minimum, one, if it is maximum). A pair of points of tangency of $\mathbb{R}A$ with lines from $Z$ is said to be switching, if the points of the pair has distinct indices and one of the points is positive while the other one is negative; otherwise the pair is said to be inessential. See Figure 27.

If $A$ is a nonsingular conic with $\mathbb{R}A\neq\varnothing$ and $\mathbb{R}A\subset C$ then the tangency points make a switching pair. The same is true for any convex oval. When an oval is deforming and loses its convexity, new points of tangency may appear. If the deformation is generic, the points of tangency appear and disappear pairwise. Each time appearing pair is an inessential pair of points with distinct indices. Any oval can be deformed (topologically) into a convex one. Tracing the births and deaths of points of tangency it is not difficult to prove the following lemma.

Denote the closure of $(\mathbb{C}A\smallsetminus\mathbb{R}A)\cap(\bigcup_{t\in I}\mathbb{C}L_{t})$ by $S$. Fix one of the decomposions into pairs of the set of points of tangency of lines from $Z$ with each component of $\mathbb{R}A\cap C$ existing by 4.2.B. By a chain of points of tangency call a sequence of points of tangency, in which any two consecutive points either belong to one of selected pairs or lie in the same component of $S$. A sequence consisting of ovals, on which the selected switching pairs of points of tangency from the chain lie, is called a chain of ovals. Thus the set of ovals of $A$ lying in $C$ appeared to be decomposed to chains of ovals. The next theorem follows obviously from 4.2.A.

The next theorem follows in an obvious way from 4.2.C. Contrary to the previous one, it deals with the signs of ovals with respect to the one-sided component in the case of odd degree and outer ovals in the case of even degree.

In proofs of 3.8.G, 3.8.H and 3.8.I, the theory developed in the previous section can be replaced by the following Theorem 4.3.A. Although this theorem can be obtained as a corollary of Theorem 4.2.C, it is derived here from Theorem 2.3.A and the complex orientation formula, and in the proof no chain of ovals is used. The idea of this approach to Fiedler’s alternation of orientations is due to V. A. Rokhlin.

In this section Theorem 3.8.G is proved, i.e. it is proved that there is no nonsingular curve of degree 7 with real scheme $\langle J\amalg\,1\langle 14\rangle\rangle$.

Assume the contrary: suppose that there exists a nonsingular curve $X$ of degree 7 with real scheme $\langle J\amalg\,1\langle 14\rangle\rangle$.

Being an M-curve, $X$ is of type I (see 2.6.B) and, hence, has a complex orientation.

The next ingredient in the proof of Theorem 3.8.G is a kind of convexity in disposition of interior ovals. Although we study a projective problem, it is possible to speak about convexity, if it is applied to interior ovals. The exact sense of this convexity is provided in the following statement.

Our consideration of real algebraic surfaces will be based on definitions similar to the definitions that we used in the case of curves. In particular, by a real algebraic surface of degree $m$ in the 3-dimensional projective space we shall mean a real homogeneous polynomial of degree $m$ in four variables considered up to a constant factor.

Obvious changes adapt definitions of sets of real and complex points, singular points, singular and nonsingular curves and rigid isotopy to the case of surfaces in $\mathbb{R}P^{3}$. Exactly as in the case of curves one formulates the topological classification problem (cf. 1.1.A above):

However, the isotopy classification problem 1.1.B splits into two problems:

The reason for this splitting is that, contrary to the case of projective plane, there exists a homeomorphism of $\mathbb{R}P^{3}$ non-isotopic to the identity. Indeed, 3-dimensional projective space is orientable, and the mirror reflection of this space in a plane reverses orientation. Thus the reflection is not isotopic to the identity. However, there are only two isotopy classes of homeomorphisms of $\mathbb{R}P^{3}$. It means that the difference between 5.1.B and 5.1.C is not really big. Although the isotopy classification problem is finer, to resolve it, one should add to a solution of the ambient topological classification problem an answer to the following question:

Each of these problems has been solved only for $m\leq 4$. The difference between 5.1.B and 5.1.C does not appear: the solutions of 5.1.B and 5.1.C coincide with each other for $m\leq 4$. (Thus Problem 5.1.D has a simple answer for $m\leq 4$: any nonsingular real algebraic surface of degree $\leq 4$ is isotopic to its mirror image.) For $m\leq 3$ solutions of 5.1.A and 5.1.B also coincide, but for $m=4$ they are different: there exist nonsingular surfaces of degree 4 in $\mathbb{R}P^{3}$ which are homeomorphic, but embedded in $\mathbb{R}P^{3}$ in a such a way that there is no homeomorphism of $\mathbb{R}P^{3}$ mapping one of them to another. The simplest example is provided by torus defined by equation

$(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+3x_{0}^{2})^{2}-16(x_{1}^{2}+x_{2}^{2})x_{0}^{2 }=0$

and the union of one-sheeted hyperboloid and an imaginary quadric (perturbed, if you wish to have a surface without singular points even in the complex domain)

Similar splitting happens with the rigid isotopy classification problem. Certainly, it may be transferred literally:

However, since there exists a projective transformation of $\mathbb{R}P^{3}$, which is not isotopic to the identity (e.g., the mirror reflection in a plane) and a real algebraic surface can be nonisotopic rigidly to its mirror image, one may consider the following rougher problem:

Again, as in the case of topological isotopy and homeomorphism problem, the difference between these two problems is an amphichirality problem:

Problems 5.1.E, 5.1.F and 5.1.G have been solved also for $m\leq 4$. For $m\leq 3$ the solutions of 5.1.E and 5.1.F coincide with each other and with the solutions of 5.1.A, 5.1.B and 5.1.C. For $m\leq 2$ all these problems belong to the traditional analytic geometry. The solutions are well-known and can be found in traditional textbooks on analytic geometry. The case $m=3$ is also elementary. It was studied in the nineteenth century. The solution is associated with names of Schläfli and Klein. The case $m=4$ is really difficult. Although the first attempts of a serious attack were undertaken in the nineteenth century, too, and among the attackers we see D. Hilbert and K. Rohn, the complete solutions of all classification problems listed above were obtained only in the seventies and eighties. Below, in Subsection LABEL:s5.5n, I will discuss the results and methods. In higher degrees even the most rough problems, like the Harnack problem on the maximal number of components of a surface of degree $m$ are still open.

For brevity, we shall refer to closed two-dimensional submanifolds of $\mathbb{R}P^{3}$ as topological spatial surfaces, or simply surfaces when there is no danger of confusion.

Since the homology group $H_{2}(\mathbb{R}P^{3};\,\mathbb{Z}_{2})$ is $\mathbb{Z}_{2}$, a connected surface can be situated in $\mathbb{R}P^{3}$ in two ways: zero-homologous, and realizing the nontrivial homology class.

In the first case it divides the projective space into two domains being the boundary for both domains. Hence, the surface divides its tubular neighborhood, i. e. it is two-sided.

In the second case the complement of the surface in the projective space is connected. (If it was not connected, the surface would bound and thereby realize the zero homology class.) Moreover, it is one-sided.

The latter can be proved in many ways. For example, if the surface was two-sided and its complement was connected, there would exist a nontrivial infinite cyclic covering of $\mathbb{R}P^{3}$, which would contradict the fact that $\pi_{1}(\mathbb{R}P^{3})=\mathbb{Z}_{2}.$ The infinite cyclic covering could be constructed by gluing an infinite sequence of copies of $\mathbb{R}P^{3}$ cut along the surface: each copy has to be glued along one of the sides of the cut to the other side of the cut in the next copy.

Another proof: take a projective plane, make it transversal to the surface, and consider the curve which is their intersection. Its homology class in $\mathbb{R}P^{2}$ is the image of the nontrivial element of $H_{2}(\mathbb{R}P^{3};\,\mathbb{Z}_{2})$ under the inverse Hopf homomorphism $in^{!}:H_{2}(\mathbb{R}P^{3};\,\mathbb{Z}_{2})\to H_{1}(\mathbb{R}P^{2};\, \mathbb{Z}_{2})$. This is an isomorphism, as one can see taking the same construction in the case when the surface is another projective plane. Thus the intersection is a one-sided curve in $\mathbb{R}P^{2}$. Hence the normal fibration of the original surface in $\mathbb{R}P^{3}$ is not trivial. This means that the surface is one-sided.

A connected surface two-sidedly embedded in $\mathbb{R}P^{3}$ is orientable, since it bounds a part of the ambient space which is orientable. Therefore, such a surface is homeomorphic to sphere or to sphere with handles. There is no restriction to the number of handles: one can take an embedded sphere bounding a small ball, and adjoin to it any number of handles.

A one-sidedly embedded surface is nonorientable. Indeed, its normal bundle is nonorientable, while the restriction of the tangent bundle of $\mathbb{R}P^{3}$ to the surface is orientable (since $\mathbb{R}P^{3}$ is). The restriction of the tangent bundle of $\mathbb{R}P^{3}$ to the surface is the Whitney sum of the normal and tangent bundles of the surface. Therefore it cannot happen that only one of these three bundles is not orientable.

Contrary to the case of two-sided surfaces, in the case of one-sided surfaces there is an additional restriction on their topological types.

In particular, it is impossible to embed a Klein bottle to $\mathbb{R}P^{3}$. (The Euler characteristic of a connected surface two-sidedly embedded into $\mathbb{R}P^{3}$ is even, but it follows from orientability: the Euler characteristic of any closed oriented surface is even.) By topological classification of closed surfaces, a nonorientable connected surface with odd Euler characteristic is homeomorphic to the projective plane or to the projective plane with handles. Any surface of this sort can be embedded into $\mathbb{R}P^{3}$: for the projective plane $\mathbb{R}P^{3}$ is the native ambient space, and one can adjoin to it in $\mathbb{R}P^{3}$ any number of handles. We denote a sphere with $g$ handles by $S_{g}$ and a projective plane with $g$ handles by $P_{g}$.

A one-sided connected surface in $\mathbb{R}P^{3}$ contains a loop which is not contractible in $\mathbb{R}P^{3}$. Such a loop can be detected in the following way: Consider the intersection of the surface with any one-sided transversal surface (e. g., $\mathbb{R}P^{2}$ or a surface obtained from the original one by a small shift). The homology class of the intersection curve is the self-intersection of the nonzero element of $H_{2}(\mathbb{R}P^{3}\,;\,\mathbb{Z}_{2})$. Since the self-intersection is the nonzero element of $H_{1}(\mathbb{R}P^{3}\,;\,\mathbb{Z}_{2})$, the intersection curve contains a component noncontractible in $\mathbb{R}P^{3}$.

A two-sided connected surface in $\mathbb{R}P^{3}$ can contain no loops noncontractible in $\mathbb{R}P^{3}$ (this happens, for instance, if the surface lies in an affine part of $\mathbb{R}P^{3}$). Of course, if a surface contains a loop noncontractible in $\mathbb{R}P^{3}$, it is not contractible in $\mathbb{R}P^{3}$ itself. Moreover, then it meets any one-sided surface, since the noncontractible loop realizes the nonzero element of $H_{1}(\mathbb{R}P^{3}\,;\,\mathbb{Z}_{2})$ and this element has nonzero intersection number with the homology class realized by a one-sided surface.

If any loop on a connected surface $S$ embedded in $\mathbb{R}P^{3}$ is contractible in $\mathbb{R}P^{3}$ (which means that the embedding homomorphism $\pi_{1}(S)\to\pi_{1}(\mathbb{R}P^{3})$ is trivial), then there is no obstruction to contract the embedding, i. e., to construct a homotopy between the embedding $S\to\mathbb{R}P^{3}$ and a constant map. One can take a cell decomposition of $S$, contract the 1-skeleton (extending the homotopy to the whole $S$), and then contract the map of the 2-cell, which is possible, since $\pi_{2}(\mathbb{R}P^{3})=0$. A surface of this sort is called contractible (in $\mathbb{R}P^{3}$).

It may happen, however, that there is no isotopy relating the embedding of a contractible surface with a map to an affine part of $\mathbb{R}P^{3}$. The simplest example of a contractible torus which cannot be moved by an isotopy to an affine part of $\mathbb{R}P^{3}$ is shown in Figure 32.

As it was stated above, the complement $\mathbb{R}P^{3}\smallsetminus S$ of a connected surface $S$ two-sidedly embedded in $\mathbb{R}P^{3}$ consists of two connected components. If $S$ is not contractible in $\mathbb{R}P^{3}$ then both of them are not contractible, since a loop on $S$ noncontractible in $\mathbb{R}P^{3}$ can be pushed to each of the components. They may be positioned in $\mathbb{R}P^{3}$ in the same way.

The simplest example of this situation is provided by a one-sheeted hyperboloid. It is homeomorphic to torus and its complement consists of two solid tori. So, this is a Heegaard decomposition of $\mathbb{R}P^{3}$. There exists an isotopy of $\mathbb{R}P^{3}$ made of projective transformation exchanging the components.  $(^{3})$

A connected surface decomposing $\mathbb{R}P^{3}$ into two handlebodies is called a Heegaard surface. Heegaard surfaces are the most unknotted surfaces among two-sided noncontractible connected surfaces. They may be thought of as unknotted noncontractible surfaces.

If a connected surface $S$ is contractible in $\mathbb{R}P^{3}$, then the components $C_{1}$ and $C_{2}$ can be distinguished in the following way: for one of them, say $C_{1}$, the inclusion homomorphism $\pi_{1}(C_{1})\to\pi_{1}(\mathbb{R}P^{3})$ is trivial, while for the other one the inclusion homomorphism $\pi_{1}(C_{2})\to\pi_{1}(\mathbb{R}P^{3})$ is surjective. This follows from the van Kampen theorem. The component with trivial homomorphism is called the interior of the surface. It is contractible in $\mathbb{R}P^{3}$ (in the same sense as the surface is).

A contractible connected surface $S$ in $\mathbb{R}P^{3}$ is said to be unknotted, if it is contained in some ball $B$ embedded into $\mathbb{R}P^{3}$ and divides this ball into a ball with handles (which is the interior of $S$) and a ball with handles with an open ball deleted. Any two unknotted contractible surfaces of the same genus are ambiently isotopic in $\mathbb{R}P^{3}$. Indeed, first the balls containing them can be identified by an ambient isotopy (see, e. g., Hirsch Hir-76, Section 8.3), then it follows from uniqueness of Heegaard decomposition of sphere that there is an orientation preserving homeomorphism of the ball mapping one of the surfaces to the other. Any orientation preserving homeomorphism of a 3-ball is isotopic to the identity.

At most one component of a (closed) surface embedded in $\mathbb{R}P^{3}$ may be one-sided. Indeed, a one-sided closed surface cannot be zero-homologous in $\mathbb{R}P^{3}$ and the self-intersection of its homology class (which is the only nontrivial element of $H_{2}(\mathbb{R}P^{3}\,;\,\mathbb{Z}_{2})$) is the nonzero element of $H_{1}(\mathbb{R}P^{3}\,;\,\mathbb{Z}_{2})$. Therefore any two one-sided surfaces in $\mathbb{R}P^{3}$ intersect.

Moreover, if an embedded surface has a one-sided component, then all other components are contractible. The contractible components are naturally ordered: a contractible component of a surface can contain other contractible component in its interior and this gives rise to a partial order in the set of contractible components. If the interior of contractible surface $A$ contains a surface $B$, then one says that $A$ envelopes $B$.

The connected components of a surface embedded in $\mathbb{R}P^{3}$ divide $\mathbb{R}P^{3}$ into connected regions. Let us construct a graph of adjacency of these regions: assign a vertex to each of the regions and connect two regions with an edge if the corresponding regions are adjacent to the same connected two-sided component of the surface. Since the projective space is connected and its fundamental group is finite, the graph is contractible, i. e., it is a tree. It is called region tree of the surface.

Consider now a (closed) surface without one-sided components. It may contain several noncontractible components. They decompose the projective space into connected domains, each of which is not contractible in $\mathbb{R}P^{3}$. Let us construct a graph of adjacency of these domains: assign a vertex to each of the domains and connect two vertices with an edge if the corresponding domains are adjacent. Edges of the graph correspond to noncontractible components of the surface. For the same reasons as above, this graph is contractible, i. e. it is a tree. This tree is called the domain tree of the surface.

Contractible components of the surface are distributed in the domains. Contractible components which are contained in different domains cannot envelope one another. Contractible components of the surface which lie in the same domain are partially ordered by enveloping. They divide the domain into regions. Each domain contains only one region which is not contractible in $\mathbb{R}P^{3}$. If the domain does not coincide with the whole $\mathbb{R}P^{3}$ (i.e., the surface does contain noncontractible components), then this region can be characterized also as the only region which is adjacent to all the noncontractible components of the surface comprising the boundary of the domain. Indeed, contractible components of the surface cannot separate noncontractible ones.

The region tree of a surface contains a subtree isomorphic to the domain tree, since one can assign to each domain the unique noncontractible region contained in the domain and two domains are adjacent iff the noncontractible regions contained in them are adjacent. The complement of the noncontractible domains tree is a union of adjacency trees for contractible subdomains contained in each of the domains.

Let us summarize what can be said about topology of a spatial surface in the terms described above.

If a surface is one-sided (i. e., contains a one-sided component), then it is a disjoint sum of a projective plane with handles and several (maybe none) spheres with handles. Thus, it is homeomorphic to

$P_{g}\amalg S_{g_{1}}\amalg\dots S_{g_{k}},$

where $\amalg$ denotes disjoint sum.

All two-sided components are contractible and ordered by enveloping. The order is easy to incorporate into the notation of the topological type above. Namely, place notations for components enveloped by a component $A$ immediately after $A$ inside brackets $\langle\ \rangle$. For example,

$P_{0}\amalg S_{1}\amalg S_{1}\amalg S_{0}\langle S_{1}\amalg S_{0}\amalg S_{2} \langle S_{1}\amalg S_{0}\rangle\amalg S_{2}\langle S_{1}\amalg S_{0}\rangle\rangle$

denotes a surface consisting of a projective plane, two tori, which do not envelope any other component, a sphere, which envelopes a torus and a sphere without components inside them and a two spheres with two handles each of which envelopes empty sphere and torus. To make the notations shorter, let us agree to skip index 0, i. e. denote projective plane $P_{0}$ by $P$ and sphere $S_{0}$ by $S$, and denote the disjoint sum of $k$ fragments identical to each other by $k$ followed by the notation of the fragment. These agreements shorten the notation above to

$P\amalg 2S_{1}\amalg S\langle S_{1}\amalg S\amalg 2S_{2}\langle S_{1}\amalg S \rangle\rangle.$

If a surface is two-sided (i. e. does not contain a one-sided component), then it is a disjoint sum $S_{g_{1}}\amalg\dots S_{g_{k}},$ of spheres with handles. To distinguish in notations the components noncontractible in $\mathbb{R}P^{3}$, we equip the corresponding symbols with upper index 1. Although we do not make any difference between two components of the complement of noncontractible connected surface (and there are cases when they cannot be distinguished), in notations we proceed as if one of the components is interior: the symbols denoting components of the surface which lie in one of the components of the complement of the noncontractible component $A$ are placed immediately after the notation of $A$ inside braces $\{\ \}$. Our choice is the matter of convenience. It correspond to the well-known fact that usually, to describe a tree, one introduces a partial order on the set of its vertices.

In these notations,

$S_{1}\amalg S\langle 3S\rangle\amalg S_{1}^{1}\{S_{3}\amalg 2S_{2}^{1}\{3S \amalg S_{1}\}\}$

denotes a two-sided surface containing three noncontractible components. One of them is a torus, two others are spheres with two handles. The torus bounds a domain containing a contractible empty torus and a sphere enveloping three empty spheres. There is a domain bounded by all three noncontractible components. It contains a contractible empty sphere with three handles. Each of the noncontractible spheres with two handles bounds a domain containing empty contractible torus and three empty spheres.

This notation system is similar to notations used above to described isotopy types of curves in the projective plane. However, there is a fundamental difference: the notations for curves describe the isotopy type of a curve completely, while the notations for surfaces are far from being complete in this sense. Although topological type of the surface is described, knotting and linking of handles are completely ignored. In the case when there is no handle, the notation above does provide a complete description of isotopy type.

As in the case of real plane projective curves, the set of real points of a nonsingular spatial surface of degree $m$ is one-sided, if $m$ is odd, and two-sided, if $m$ is even. Indeed, by the Bézout theorem a generic line meets a surface of degree $m$ in a number of points congruent to $m$ modulo 2. On the other hand, whether a topological surface embedded in $\mathbb{R}P^{3}$ is one-sided or two-sided, can be detected by its intersection number modulo 2 with a generic line: a surface is one-sided, iff its intersection number with a generic line is odd.

There are some other restrictions on topology of a nonsingular surface of degree $m$ which can be deduced from the Bézout theorem.

The proof is a straightforward generalization of the proof of 5.3.D.$\square$

Surprisingly, Bézout theorem gave much less restrictions in the case of surfaces than in the case of plane curves. In particular, it does not give anything like Harnack Inequality. Most of restrictions on topology of surfaces are analogous to the restrictions on flexible curves and were obtained using the same topological tools. Here is a list of the restrictions, though it is non-complete in any sense.

The restrictions are formulated below for a nonsingular real algebraic surface $A$ of degree $m$ in the 3-dimensional projective space. In these formulations and in what follows we shall denote the $i$-th Betti number of $X$ over field $\mathbb{Z}_{2}$ (which is nothing but $\dim_{\mathbb{Z}_{2}}H_{i}(X\,;\,\mathbb{Z}_{2})$) by $b_{i}(X)$. In particular, $b_{0}(X)$ is the number of components of $X$. By $b_{*}(X)$ we denote the total Betti number, i. e. $\sum_{i=0}^{infty}b_{i}(X)=\dim_{\mathbb{Z}_{2}}H_{*}(X\,;\,\mathbb{Z}_{2})$.

Denote the numbers of orientable components of $\mathbb{R}A$ with positive, zero and negative Euler characteristic by $k^{+}$, $k^{0}$ and $k^{-}$ respectively.

Surfaces of degree 1 and 2 are well-known. Any surface of degree 1 is a projective plane. All of them are transformed to each other by a rigid isotopy consisting of projective transformations of the whole ambient space $\mathbb{R}P^{3}$.

Nonsingular surfaces of degree 2 (nonsingular quadrics) are of three types. It follows from the well-known classification of real nondegenerate quadratic forms in 4 variables up to linear transformation. Indeed, by this classification any such a form can be turned to one of the following:

1. (1)

   $+x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}$,

2. (2)

   $+x_{0}^{2}+x_{1}^{2}+x_{2}^{2}-x_{3}^{2}$,

3. (3)

   $+x_{0}^{2}+x_{1}^{2}-x_{2}^{2}-x_{3}^{2}$,

4. (4)

   $+x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}$,

5. (5)

   $-x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}$.

Multiplication by $-1$ identifies the first of them with the last and the second with the fourth reducing the number of classes to three. Since the reduction of a quadratic form to a canonical one can be done in a continuous way, all quadrics belonging to the same type also can be transformed to each other by a rigid isotopy made of projective transformations.

The first of the types consists of quadrics with empty set of real points. In traditional analytic geometry these quadrics are called imaginary ellipsoids. A canonical representative of this class is defined by equation $x_{0}^{2}+x_{1}^{2}+x_{3}^{2}+x_{4}^{2}=0$.

The second type consists of quadrics with the set of real points homeomorphic to sphere. In the notations of the previous section this is $S$. The canonical equation is $x_{0}^{2}+x_{1}^{2}+x_{2}^{2}-x_{3}^{2}=0$.

The third type consists of quadrics with the set of real points homeomorphic to torus. They are known as one-sheeted hyperboloids. The set of real points is not contractible (it contains a line), so in the notations above it should be presented as $S_{1}^{1}$. The canonical equation is $x_{0}^{2}+x_{1}^{2}-x_{2}^{2}-x_{3}^{2}=0$.

Quadrics of the last two types (i. e., quadrics with nonempty real part) can be obtained by small perturbations of a union of two real planes. To obtain a quadric with real part homeomorphic to sphere, one may perturb the union of two real planes in the following way. Let the plane be defined by equations $L_{1}(x_{0},x_{1},x_{2},x_{3})=0$ and $L_{2}(x_{0},x_{1},x_{2},x_{3})=0$. Then the union is defined by equation $L_{1}(x_{0},x_{1},x_{2},x_{3})L_{2}(x_{0},x_{1},x_{2},x_{3})=0$. Perturb this equation adding a small positive definite quadratic form. Say, take

$L_{1}(x_{0},x_{1},x_{2},x_{3})L_{2}(x_{0},x_{1},x_{2},x_{3})+\varepsilon(x_{0} ^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2})=0$

with a small $\varepsilon>0$. This equation defines a quadric. Its real part does not meet plane $L_{1}(x_{0},x_{1},x_{2},x_{3})=L_{2}(x_{0},x_{1},x_{2},x_{3})$, since on the real part of the quadric the product $L_{1}(x_{0},x_{1},x_{2},x_{3})L_{2}(x_{0},x_{1},x_{2},x_{3})$ is negative. Therefore the real part of the quadric is contractible in $\mathbb{R}P^{3}$. Since it is obtained by a perturbation of the union of two planes, it is not empty, provided $\varepsilon>0$ is small enough. As easy to see, it is not singular for small $\varepsilon>0$. Cf. Subsection LABEL:s11.5d. Of course, this can be proved explicitely, as an exercise in analytic geometry. See Figure 33

To obtain a noncontractible nonsingular quadric (one-sheeted hyperboloid), one can perturb the same equation $L_{1}(x_{0},x_{1},x_{2},x_{3})L_{2}(x_{0},x_{1},x_{2},x_{3})=0$, but by a small form which takes both positive and negative values on the intersection line of the planes. See Figure 34.

Nonsingular surfaces of degree 3 (nonsingular cubics) are of five types. Here is the complete list of there topological types:

$P,\quad P\amalg S,\quad P_{1},\quad P_{2},\quad P_{3}.$

Let us prove, first, that only topological types from this list can be realized. Since the degree is odd, a nonsingular surface has to be one-sided. By 5.3.D if it is not connected, then it is homeomorphic to $P\amalg S$. By the Generalized Harnack Inequality 5.3.G, the total Betti number of the real part is at most $3^{3}-4\times 3^{2}+6\times 3=9$. On the other hand, the first Betti number of a projective plane with $g$ handles is $1+2g$ and the total Betti number $b_{*}(P_{g})$ is $3+2g$. Therefore in the case of a nonsingular cubic with connected real part, it is of the type $P_{g}$ with $g\leq 3$.

All the five topological types are realized by small perturbations of unions of a nonsingular quadric and a plane transversal to one another. This is similar to the perturbations considered above, in the case of spatial quadrics. See Figures 35 and 36.

An alternative way to construct nonsingular surfaces of degree 3 of all the topological types is provided by a connection between nonsingular spatial cubics and plane nonsingular quartics. More precisely, there is a correspondence assigning a plane nonsingular quartic with a selected real double tangent line to a nonsingular spatial cubic with a selected real point on it. It goes as follows. Consider the projection of the cubic from a point selected on it to a plane. The projection is similar to the well-known stereographic projection of a sphere to plane.